Characterizing Mode I Fracture Behaviors of Wood Using Compact Tension in Selected System Crack Propagation

Abstract

The fracture behaviors of four wood species commonly used in wood products were characterized when subjected to compact tension (CT) load in radial-longitudinal (RL) system crack propagation. Meanwhile, the failure modes of evaluated CT samples were compared and analyzed using the fractal dimension method. The results showed that wood species had a significant effect on fracture characteristic values, including maximum fracture load, critical stress intensity factor and fracture energy. These characteristic values changed in the same way, i.e., beech wood CT samples obtained the maximum characteristic values, followed by ash, okoume, and poplar in descending order. The fracture behaviors of all wood species evaluated can be described by combining linear and exponential fitting equations at the crack initial stage and evolution stage, respectively. Linear positive proportional relationships were observed between fracture characteristic values and fractal dimensions calculated using cracks in front and left views of CT samples. However, the relationships between fracture characteristic values and fractal dimensions calculated using fracture surfaces were negative. The fractal dimensions of cracks in front view of CT samples could be a better indicator used to predict critical stress intensity factor and fracture energy, which had greater correlation coefficients beyond 0.95.

1. Introduction

As a natural composite material, wood has been widely used in engineering fields including wood constructions and wood products [1,2,3,4]. Compared with other engineering structure materials, such as steel rods and cement concrete, wood has the advantages of being environmental friendly [5,6] and having easy processing [7], good decoration [8,9,10,11,12] and an early warning system before failure [13,14]. Crack propagation in mode I fracture is main failure mode under tension [15,16,17]. Meanwhile, critical stress intensity factor has significant effect on the cutting processing of wood and other materials [18,19]. As a result, characterizing the fracture behaviors of wood materials is critically important for structure design and safety evaluation of wood constructions and wood products.

Previous studies have conducted some valuable works on the effects of factors that influencing the fracture mechanical properties of wood materials, such as temperature [20,21], testing method [22], moisture content [23,24,25,26,27], wood species [26,28,29], fracture modes [30], cracking system [30], etc. These studies provided fundamental data for characterizing wood fracture behaviors. The following literature review summarized the main conclusions of these studies on wood fracture mechanics.

In the case of temperature, the ultimate load and fracture energy were noticeably influenced by temperature above 90 °C for wood sample tested using the double cantilever beam (DCB) method under mode I loading, while no visible influence was observed in the range of 30 °C to 90 °C [21]. Meanwhile, the critical stress intensity factor decreased with the increase of temperature ranging from 20 °C to 80 °C with an increment of 20 °C when tested under a wedge splitting (WS) test method in a radial-longitudinal (RL) system [20]. The above studies also indirectly suggested that testing method had visibly effect on critical stress intensity factor. Regarding testing method, according to material types and limitation of sample processing, the following testing methods were commonly used to measure the fracture behaviors of wood: compact tension (CT), DCB, single edge notch bending (SENB), single edge notched tension (SENT), and WS testing methods. Yoshihara and Kacamura [22] studied the effects of measurement methods on the mode I critical stress intensity factor of wood through using SENB, SENT, CT and DBC testing methods, and the results showed that four typical testing methods had their own advantage to meet different sizes of wood samples and grain orientations. On moisture content, it was reported that moisture content had a negative effect on critical stress intensity factor of mode I fracture behavior using the WS testing method [23]. Zboňák et al. [31] also suggested that an increase of moisture content led to a decrease in critical stress intensity factor. However, the specific fracture energy increased with moisture content increasing [23,25]. For wood species, in previous studies on testing critical stress intensity factor, many wood species were used as subjects, including softwoods and hardwoods. The fracture behaviors of softwoods behaved in a more ductile manner, while hardwoods behaved in brittle and more linear elastic manner [28]. Concerning crack systems, taking wood as a typical orthotropic material, usually, there are six types of samples used to evaluate the fracture behaviors of wood according to crack propagation directions, including RL, tangential-longitudinal (TL), radial-tangential (RT), longitudinal-tangential (LT), longitudinal-radial (LR), and tangential-radial (TR), with the first letter meaning the tension direction and the second one indicating the crack propagation direction [32] (Figure 1). Ohuchi et al. [33] studied the critical stress intensity factor of these six types of CT samples with different loading speeds and obtained that LR and LT samples were incompatible for critical stress intensity factor tests, and loading speed had no effect on critical stress intensity factor.

The main objective of this study was to characterize fracture behaviors of four wood species commonly used in wood products in an RL crack propagation system when subjected to compact tension load. Meanwhile, the relations between critical stress intensity factor and fractal dimensions in different locations was compared and analyzed. These provide new information for wood fracture mechanical study. Specifically, the fracture characteristic values including maximum fracture load, critical stress intensity factor and fracture energy of four evaluated wood species were compared and analyzed. The fracture behaviors of these wood species were characterized using a regression method based on the traction-separation law. Meanwhile, failure modes of tested CT blocks were evaluated using fractal dimension method, and relationships of critical stress intensity factor and fracture energy versus fractal dimensions were regressed and compared. The outcomes of this study will provide some fundamental knowledge on wood fracture behaviors.

2. Materials and Methods

2.1. Wood Materials

Wood species selected in the study were ash (Fraxinus excelsior), beech (Fagus orientalis), okoume (Aucoumea klaineana), and poplar (Poplus tomemtosa) bought from a local commercial wood supplier (Nanjing, China). All wood lumber, with dimensions of 3000 × 300 × 50 mm, was stored in the laboratory for more than 12 months before preparing samples and tests. The average widths of annual rings were 3.5 (10.4), 4.2 (12.8), 2.1 (15.4), and 2.5 (14.7) mm for ash, beech, poplar, and okoume, respectively, with coefficients of variance in percentage in parentheses.

2.2. Specimens Preparation

Following procedure described the way how to prepare CT samples. Firstly, the full-size wood lumbers were machined to 200 × 30 × 12.5 mm (length × width × thickness) in the middle part of the lumber, followed by cutting to 31.25 × 30 × 12.5 mm. Afterwards, two holes with diameter of 5 mm were drilled according to ASTM E 1820 [34], and then a 19 mm long × 1 mm thick gap was machined using a miniature band saw. Finally, a 1 mm-long prefabricated crack tip was cut by a thin and shaped knife. Figure 2 shows the specific dimensions of CT samples. All samples were stored for a week in the laboratory controlled in the chamber at 25 °C with a relative humidity of 65% and then tested in the same laboratory.

2.3. Experimental Design

The effect of wood species on fracture characteristic values, maximum fracture load, critical stress intensity factor, and energy were studied considering ash, beech, okoume, and poplar with 12 replications for each. Afterwards, failure modes of all tested CT samples were scanned using a Canon scanner (9000F MarkII, Canon, Tokyo, Japan) to obtain photos of cracks in front and left views of samples, as well as photos of fracture surfaces. Finally, fractal dimensions of these photos were calculated and relationships between these fractal dimensions and fracture characteristic values were regressed.

Figure 3 shows the setup for measuring fracture characteristic values of CT samples through a 20 kN universal test machine (AGS-X, SHIMDZU, Kyoto, Japan) with a load rate of 2 mm/min. A 5 N-preload was imposed first to ensure the steel pins made full contact with the holes of the CT samples. The load and displacement were recorded by test machine automatically.

2.4. Calculating Method of Fractal Dimentions

The main steps for calculating fractal dimensions of cracks using photos of failure samples were as followings: (1) the photos of cracks in front and left views, as well as photos of fracture surfaces of CT samples were transferred to grayscale images with a resolution of 600 dpi in TIF format; then (2) the box dimension method was applied to calculate fractal dimensions of these grayscale images through using Fraclab 2.2 toolbox (INRIA, France) built in the Matlab R2014a (MathWorks, Natick, MA, USA); and (3) the specific settings were that box sizes ranged from 1/2048 to 1/2, and the relationship between box sizes and corresponding numbers of box was fitted by linear logarithmic least square regression method as shown in Equation (1). The slop, D, of the fitting line was the fractal dimension [35,36]. Figure 4 shows an example for calculating fractal dimensions, which indicates that the fractal dimension is 1.332.

$${\mathrm{Log}}_2\left(N\right)=-D{\mathrm{Log}}_2\left(x\right)+C$$

(1)

where N is the number of elements corresponding to the certain element size; x is box size; C is a constant in equation.

2.5. Statistical Analysis

The effects of wood species on critical stress intensity factor and fracture energy under compact tension test in RL crack propagation system were analyzed using analysis of variance (ANOVA) general linear model (GLM) procedure. Mean comparisons using the protected least significant difference (LSD) multiple comparison procedure was conducted if any significant was identified. All these analyses were performed at 5% significance level using SPSS 22.0 (IBM, Armonk, NY, USA). In addition, linear regression analysis and data plots were conducted using Origin 9.10 (OriginLab Corporation, Northampton, UK).

3. Results and Discussion

3.1. Density and Moisture Content

Density and moisture content of four wood species evaluated were measured according to ASTM D 2395 [37] and ASTM D 4442 [38], respectively, after compact tension tests, and the results were shown in

Table 1. Density and moisture content of four wood species evaluated.

Wood Species	Beech	Ash	Okoume	Poplar
MC (%)	9.22 (2.7)	9.69 (2.4)	9.40 (2.0)	8.64 (2.5)
Density (g/cm3)	0.68 (5.5)	0.74 (2.5)	4.76 (2.8)	0.45 (9.2)

Note: values in parenthesis are coefficient of variance (COV) in percentage.

.

3.2. Fracture Characteristics

Figure 5 shows typical load-displacement curves in an RL system CT test of wood species evaluated in this work, which indicates that these curves experienced a linear elastic stage at initial stage before fracture, and then underwent a nonlinear stage at the fracture evolution stage. The maximum load was obtained at the initial crack point. Critical stress intensity factor could be calculated by Equation (2). For wood, previous studies have defined $f\left(\frac{a}{W}\right)$ as shown in the following polynomial Equation (3) [33,39,40,41].

$$K_{IC}=\frac{P}{B\sqrt{W}}f\left(\frac{a}{W}\right)$$

(2)

The polynomial function is applicable to the orthotropic materials when $0.3\le \frac{a}{W}\le 0.7$ [41,42]. In this study, a, B and W are equal to 12.5 mm, 12.5 mm, and 25 mm, respectively, and $f\left(\frac{a}{W}\right)$ is equal to 8.67, calculated by Equation (3).

$$f\left(\frac{a}{W}\right)=29.6{\left(\frac{a}{W}\right)}^{1/2}-185.5{\left(\frac{a}{W}\right)}^{3/2}+655.7{\left(\frac{a}{W}\right)}^{5/2}-1017.0{\left(\frac{a}{W}\right)}^{7/2}+638.9{\left(\frac{a}{W}\right)}^{9/2}$$

(3)

where K_IC refers to the critical stress intensity factor ($\mathrm{MPa}\sqrt{\mathrm{m}}$); P is maximum load value (N); B infers to thickness of the specimen (m); W is the width of the specimen (m); a refers to the length from the center of the hole to the crack point (m); and $f\left(\frac{a}{W}\right)$ is geometry function.

In order to further study the fracture mechanical behaviors of wood when subjected to CT load. The traction separation law was adopted to describe the fracture behavior of wood CT blocks evaluated in this study. Figure 6 shows a typical traction-separation curve of beech wood CT blocks fractured in RL system compact tension. In initial stage, the linear elastic mechanical behavior was observed, and linear fit line, y_L, was regressed. In fracture evolution stage, the nonlinear mechanical behavior was fitted by exponential function, y_E. In addition, when the traction stress reached its maximum value, the fracture occurred immediately. The maximum stress was calculated using maximum fracture force divided by normal surface of CT block. Equation (4) was used to calculate fracture energy [43]; in other words, according to geometric meaning of traction-separation curve, the area of shaded region in fracture evolution stage was the fracture energy.

$$G={{\displaystyle \int }}_{{\delta }^o}^{{\delta }^f}{y}_{\mathrm{E}}{d}_{\delta }$$

(4)

where G is fracture energy in J/mm²; δ^o and δ^f are separation distance at initial fracture point and complete failure point; and y_E is exponential fit equation in the fracture evolution stage.

Table 2. Summaries of compact tension fracture characteristic values.

Wood Species	Maximum Fracture Load (N)	Critical Stress Intensity Factor $\left(\mathbf{MPa}\sqrt{\mathbf{m}}\right)$	Maximum Stress (MPa)	Fracture Energy (J/mm2)
Ash	133.5 (18.3) B	0.649 B	0.854 B	0.148 (17.9) B
Beech	164.3 (14.5) A	0.798 A	1.050 A	0.218 (11.9) A
Okoume	87.3 (12.8) C	0.424 C	0.559 C	0.086 (18.7) C
Poplar	73.5 (23.9) C	0.357 C	0.470 C	0.064 (20.6) D

Note: values in parenthesis are coefficient of variance in percentage. The values in the same column not followed by a common letter are significantly different one from another at the 5% significance level.

summarizes the fracture characteristic values of all wood species when subjected to compact tension test, which indicates that the effect of wood species on critical stress intensity factor and fracture energy are statistically significant at 5% significance level. Meanwhile, mean comparison results show that, for maximum fracture load, critical stress intensity factor and maximum stress, beech wood’s effect is significantly higher than those of ash, followed by okoume and poplar. However, the difference between okoume and poplar is not significant at 5% significance level. In case of fracture energy, the fracture energy of beech is significantly greater than those of ash followed by okoume and poplar.

Table 3. Linear and exponential fit equations for initial and fracture evolution stages.

Wood Species	yL	RL2	yE	RE2
Ash	2.7401x − 0.03437	0.997	4.82586 × 0.00521x + 0.01468	0.9916
Beech	2.8437x − 0.06339	0.998	13.8640 × 0.00378x + 0.00108	0.9828
Okoume	1.88874x − 0.07901	0.981	5.57933 × 0.00109x + 0.00112	0.9960
Poplar	1.42949x − 0.01307	0.997	3.96315 × 0.00219x − 0.01301	0.9905

shows fitting equations and their corresponding correlation coefficients of all wood species evaluated in this study, in the initial stage, y_L, and in the fracture evolution stage, y_E. In the linear stage, the slopes of y_L dominated the differences among wood species. In the fracture evolution stage, the slopes and the bases of exponential functions represented differences among wood species, especially for the later one. Meanwhile, the correlation coefficients, R², were all very close to 1, suggesting these fitting equations were capable of describing the traction-separation curves of fracture behavior of wood CT blocks n RL crack propagation system when subjected to compact tension load.

3.3. Failure Modes and Fractal Dimensions

Figure 7 shows the typical CT failure modes of all wood species evaluated in this study in front and left views, as well as fracture surfaces. Meanwhile, the corresponding cracks photos were extracted and transferred to binary images for calculating fractal dimensions of them according to the method described in Section 2.4.

Table 4. Fractal dimensions of cracks lines and surfaces of four wood species evaluated.

Wood Species	Fractal Dimension
	Front	Left	Surface
Beech	1.3893 (8.2)	1.3142 (9.2)	1.7268 (20.6)
Ash	1.3321 (14.5)	1.3133 (11.3)	1.8202 (17.8)
Okoume	1.2950 (7.6)	1.2766 (6.4)	1.8064 (15.3)
Poplar	1.2738 (6.8)	1.2605 (7.5)	1.8234 (16.2)

Note: values in parenthesis are coefficient of variance in percentage.

summarizes the fractal dimensions of the photos of crack lines and fracture surfaces in mode I crack propagation in RL system under compact tension load, which indicates that the fractal dimensions of cracks in front and left views of CT samples, in descending order are beech, ash, okoume, and poplar. Meanwhile, the fractal dimensions of the cracks in front views of all evaluated wood species were greater than those of the cracks in left views. However, the fractal dimensions of fracture surfaces were not in the same trend. This could have resulted from the error generated during binary transfer of photos, since the fracture surfaces were too complex to obtain accurate binary photos. Therefore, the photo of the fracture surface is incompatible for fractal dimension calculation.

Figure 8 shows individual data points of critical stress intensity factor, and fracture energy versus fractal dimensions of cracks in front view, left view, and surface of all tested CT samples. It can be found that linear relationship was existed between critical stress intensity factor and fracture energy vs. fractal dimension. Therefore, a linear regression, Equation (5), was applied to fit individual data point using the least square regression method for each plotted data point.

$$y=a+bx.$$

(5)

where y is the fracture characteristic values; a and b are fitting constants.

Linear regression analysis results in four equations for fracture characteristics (critical stress intensity factor, and fracture energy) versus fractal dimensions. The fitting constants, a and b, and correlation coefficients R² were shown in

Table 5. Summaries of constants of equations for each of fitting lines of fracture characteristics versus fractal dimension of cracks.

Fracture Characteristics	Fractal Dimension of Crack	Linear Regression
		a	b	R2
Critical stress intensity factor	Front	−4.82059	4.05886	0.95708
	Left	−8.86452	7.29028	0.88654
	Surface	7.52128	−3.90728	0.42762
Fracture energy	Front	−1.6631	1.35448	0.99116
	Left	−2.7936	2.26337	0.81466
	Surface	2.72567	−1.45434	0.62554

. Analysis suggested that positive proportional relationships existed between critical stress intensity factor and fracture energy vs. fractal dimension of cracks in front and left views, while there was a negative correlation between critical stress intensity factor and fracture energy versus fractal dimensions of fracture surface. In addition, the correlation coefficients of fitting lines using fractal dimensions of cracks in front view of CT samples were much greater than those fitted using fractal dimensions of cracks in left view and fracture surface. This indicated that the fractal dimensions of cracks in the front view of CT samples could be a better indicator used to predict critical stress intensity factor and fracture energy.

4. Conclusions

In this study, the fracture behaviors of four wood species commonly used in wood products were characterized using compact tension testing in an RL system crack propagation and fractal dimension method. Following conclusions were drawn:

(1)

Wood species had significant effects on characteristic values, maximum fracture load, critical stress intensity factor and fracture energy, at 5% significance level. Beech got greater characteristic values than the others, followed by ash, okoume, and poplar in descending order.

(2)

The fracture behavior of CT wood blocks subjected to compact tension load in the RL crack propagation system could be characterized by combining linear and exponential equations in the initial stage and evolution stage, respectively, with determination coefficient greater than 0.98.

(3)

Linear positive proportional relationships were found between critical stress intensity factor and fracture energy versus fractal dimensions of cracks in front and left views of wood samples in RL crack propagation system when subjected to compact tension load. While the relationships between critical stress intensity factor and fracture energy versus fractal dimensions of fracture surfaces were negative.

(4)

The fractal dimension of cracks in front view of CT samples could be a better indicator used to predict critical stress intensity factor and fracture energy of CT blocks subjected to compact tension load with correlation coefficients greater than 0.95.

In conclusion, the fracture behaviors of wood in RL crack propagation system subjected to compact tension load can be characterized using a combination of linear and exponential fitting equations. To some degree, fractal dimensions of cracks could be capable of describing failure modes and predicting critical stress intensity factor and fracture energy of CT samples, but relationships between fracture mechanics and fractal dimension should be further investigated.